Ice cream and orbifold Riemann-Roch

TL;DR

This paper derives a closed-form orbifold Riemann-Roch formula for Hilbert series of certain Gorenstein polarized varieties with isolated orbifold points, facilitating computational applications in algebraic geometry.

Contribution

It introduces a new sum-of-parts formula for Hilbert series, called 'ice cream functions', applicable to quasismooth polarized varieties with orbifold points.

Findings

The formula is integral and Gorenstein symmetric.

Illustrations include examples of K3 surfaces and Calabi-Yau 3-folds.

The approach simplifies computer algebra computations.

Abstract

We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold X,D, under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called "ice cream functions". This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [A. Buckley and B. Szendroi, Orbifold Riemann-Roch for 3-folds with an application to Calabi-Yau geometry, J. Algebraic Geometry 14 (2005) 601--622] and [Shengtian Zhou, Orbifold Riemann-Roch and Hilbert series, University of Warwick PhD thesis, March 2011, 91+vii pp.], although the correct statements are considerably trickier. We expect to return to this in future publications.